How do "activist" and "originalist" judges conflict with each other? What would be their reasoning in the Glossip v. Gross case?

To be precise, judicial activism and originalism do not necessarily conflict with one another. Judicial activism holds that judges should not be reticent in striking down laws when they think those laws are unconstitutional. They should not be deferential to the elected branches. The opposite of judicial activism is judicial restraint. Originalism holds that judges should (when interpreting the Constitution) be bound by the original intent of the Framers. A judge can be both an originalist and an activist. It is true that judges who believe in originalism often believe in judicial restraint as well, but this is not necessarily the case. So, judicial activism and originalism do not really conflict. Activism has to do with the relationship of the judiciary to the elected branches while originalism has to do with how one interprets the Constitution.

In the Glossip case, an activist judge would not be hesitant to overrule the state legislature if he or she believed that the use of this particular combination of three drugs to execute criminals would constitute cruel and unusual punishment. An activist judge is not shy about substituting their judgement for that of the elected branches. By contrast, a person who believed in judicial restraint would do with the judgement of the state legislature that passed the law mandating this execution protocol unless they were absolutely sure the protocol violated the Constitution.

Many activist judges do not believe in originalism. They believe that judges should try to determine what values are expressed in the Constitution and should apply those values to the cases in front of them. They should not try to determine what the Framers would have felt about a given case. Instead, they should try to understand what the Constitution should mean today. Such a judge would look at Glossip and try to decide whether society has moved past capital punishment. They would not try to figure out what the Framers thought about capital punishment. Instead, they should try to determine whether that punishment is outdated by today’s standards

An originalist judge would likely rule in the way that the Supreme Court’s main originalist, Justice Scalia, did in this case. In Glossip, Scalia acted as both an originalist and a believer in judicial restraint. In order to rule on the case, Scalia asked what the Framers of the Constitution felt about the death penalty. He judged that they were split on the issue, which meant that there was no way to discern the original intent of the Framers with respect to capital punishment or to the use of these particular three drugs. Because of this, he said, the Framers left it to the people to decide whether capital punishment should be legal. Scalia believed that the Supreme Court should do the same. This is a classic statement of judicial restraint and of originalism.